Princess Luna is the younger sister of Princess Celestia who, one thousand years before the events of My Little Pony Friendship is Magic, was consumed by jealousy, turning her into Nightmare Moon. She attempted to cover Equestria in everlasting night, forcing Celestia to seal her away into the moon. Making her return in the first episode, Luna is purified by the Mane Six with the Elements of Harmony, restoring her to normal and allowing her to make amends with her sister. Despite only being featured in the first two episodes of season one, she garnered a substantial fan following with wildly variant interpretations of her personality. Subsequently, starting with season two's Luna Eclipsed, she appears sporting a design different both from Nightmare Moon and from her short appearance as Luna in Friendship is Magic, part 2 in numerous episodes, as well as the My Little Pony Equestria Girls series of movies. She is highly popular among the fandom, outranking her sister in artwork and fan fiction.

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Luna memes and alternate personalities

Humanized/anthropomorphized

Like Celestia, Luna in humanized and anthropomorphized art is typically depicted as tall and goddess-like, although Luna herself is usually depicted as being slimmer with a more elegant build to reflect her status as the younger sister. Alternate humanized/anthropomorphized fanart variations include depicting Luna as a teenager in contrast to the usually adult Celestia.

Princess Balloona

A fan term for humanized/anthropomorphized fanart that depicts Luna as being exceptionally voluptuous.

Woona

An example of "Woona"; fan art by MoongazePonies

Luna's in-show reappearance in Luna Eclipsed did not just lock in her "canon" personality, it also changed her outward appearance to more closely resemble Princess Celestia with a flowing, multi-shaded mane and a significantly larger body. One widely-disseminated comic attempts to explain the discrepancy with a mixup on the palace shower.
Luna's smaller, original design remains popular among fans. A common subject in fan art that uses the light blue single-color mane is Luna as a filly, from the time before her rebellion against Celestia. This incarnation is frequently called "Woona", a baby talk transformation of her name; the term had come to refer to Luna as a child since at least summer of 2011. Fan art of an older Luna who retains her first season design (as opposed to adopting the one from Luna Eclipsed) is also occasionally tagged as "Woona".[FA 1][FA 2].

Nightmare Moon

Nightmare Moon, as depicted by Celestialess

In the show Nightmare Moon only appears in five episodes. She is the main antagonist in the first two and is simply a prank in Luna Eclipsed. Nightmare Moon also appears in the two-parter season four premiere during Twilight's flashbacks of Nightmare Moon's banishment. After being defeated by the mane six in Friendship is Magic, part 2 she becomes Princess Luna and goes to rule with Celestia again.
In fan fiction, Nightmare Moon mostly appears as a flashback character, and only occasionally as a returning villain. There are a handful of alternative-universe fics that reinterpret the show's pilot to make Nightmare Moon win; one example is Nightmares Don't Last Forever, where Nightmare Moon develops a relationship with Twilight. Or After That Fateful Night where she becomes the Queen/Tyrant of the land.

In the story Inscape, Nightmare Moon is depicted as a parasitic entity that feeds off of and warps the desires of her host, leading to said host obsessing over what they want most.

Grown-up Luna

Prior to season 2's episode 4, fans had nothing to base Luna's character on except for a couple lines in season 1's episode 2 where she apologizes to Celestia for her Nightmare Moon tantrum. This led to a variety of guesses as to who this smaller alicorn really was and why she was apparently too busy to visit Ponyville again during season 1. The following are some of the more common or popular character traits:

Studying

Luna had been locked in the moon for 1000 years, and Equestrian culture had marched on without her. As a result, Luna needs to spend most of her free time reading up on history, law, and other subjects that have been significantly changed over the past millennium.

Socially awkward

Again, 1000 years of exile isn't conducive to being able to interact with others. While few were able to correctly predict Luna's in-show behavior, many fans rallied around the idea that Luna had become shy and unsure of herself and needed assistance, whether from the Mane 6 or members of the castle staff, to learn how to step out and take risks.

For better or worse, this seems to be a dominant part of Luna's in-show characterization.

Trolluna

In more lighthearted depictions, Luna is seen as being a prankster in a similar vein to her sister Trollestia. The sisters may get into prank wars that send the staff and guards running for cover, or conspire to lighten the boring court sessions with strange happenings.

A rather popular prank-war they got into involves Luna playing Super Mario Bros. 3, and Celestia uses her magic to cause the sun to attack Mario. Luna, simply put, is not amused.[FA 3]

She gets back at Celestia by manipulating the moon while Celestia plays The Legend of Zelda: Majora's Mask.[FA 4]

Reluctant co-regent

For one reason or another, Luna does not feel fit to take back her role in raising the moon and managing the night-time affairs of Canterlot, or feels that her contributions aren't given the same weight as her sister's. Celestia will either give her an encouraging speech and opportunities to take charge of smaller things, or simply declare a need for a vacation and drop her sister into the throne for a week.

Confident

This Luna has gotten over most of the big issues and has the gumption to take matters into her own hooves to solve the minor ones. She's also confident enough to take over Celestia's job in times of crisis, or simply because she wants it. In a negative light, this trait equates to Luna remaining jealous of Celestia and being willing to stage another coup.

The New Lunar Republic

The New Lunar Republic is a slogan and common fan labor meme centered around Luna forming a resistance movement against a tyrannical Celestia and her "Solar Empire". Depending on the story, this rebellion was the real cause of Luna's banishment, and Luna secretly carries on plotting against her sister after her return; or it is a new development, based on old resentment. The meme manifests itself in art, fan fiction, and music.